Unique Differences in Symmetric Subsets of ${\mathbb{F}}_p$

Let $p$ be a prime and let $A$ be a subset of ${\mathbb{F}}_p$ with $A=-A$ and $\mathrm{\mid }A\setminus \{0\}\mathrm{\mid }\le 2{\log }_3(p)$. Then there is an element of ${\mathbb{F}}_p$ which has a unique representation as a difference of two elements of $A$.